# Solved – Prove the consistency of estimator

Let $$y_1, dots,y_n$$ be i.i.d. random variables from $$Y$$ with density $$p(y;theta)=theta(1-y)^{theta-1}, theta > 0, y in(0,1)$$

and $$T=-ln(1-Y)sim Exp(theta)$$ with $$theta$$ rate parameter.

The log-likelihood is $$l(theta)=n ln(theta)+sumln(1-y)$$ and the MLE is $$hat theta=-frac{n}{sum ln(1-y)}$$.

I need to prove the (weak) consistency of $$hat theta$$. A sufficient condition is
$$begin{cases} lim_{n to infty}mathbb{E}(hat theta)=theta \ lim_{n to infty}mathbb{Var}(hat theta)=0 end{cases}$$

Based on MLE invariance I could write $$hat theta=frac{n}{sum t}$$, so $$lim_{n to infty}mathbb{E}(hat theta)=lim_{n to infty}mathbb{E}left(frac{n}{sum t}right)=lim_{n to infty}left(nfrac{1}{nfrac{1}{theta}}right)=theta$$

and $$mathbb{Var}(hat theta)=mathbb{Var}left(frac{n}{sum t}right)=n^2mathbb{Var}left(frac{1}{summathbb{Var}(t)}right)=n^2frac{1}{frac{n}{theta^2}}=ntheta^2$$

What am I doing wrong?

Edit with solution

My solution: if $$sum t_i sim Gamma(n,theta)$$ then $$frac{1}{sum t_i}sim IG(n,theta)$$. Now, the variance of $$IG$$ should be $$mathbb{Var}(IG(n,theta))=frac{theta^{2}}{(n-1)^2(n-2)}$$ so $$lim_{n to infty}n^2frac{theta^{2}}{(n-1)^2(n-2)}=0$$ and this prove the consistency of $$hat theta$$.

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